# Distance between the solutions of a polynomial

I have a polynomial of degree six: $x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0=0$ for which I know it will always have only two real roots and 4 complex. The coefficients $a_5\ldots{a}_0$ will change, creating a family of this kind of polynomials. I can find their roots numerically (using Matlab).

Is there a way to evaluate the distance between the real roots (in an analytical form) ? I want to prove that this distance cannot be higher than a threshold. Thank you.

• I assume you want a threshold that depends on the coefficients, else you can make the difference between the real roots arbitrarily large: Take the polynomial $x(x-R)(x^2+1)^2$. – Ethan Bolker Mar 21 '18 at 13:27
• See en.wikipedia.org/wiki/… and following. – B. Goddard Mar 21 '18 at 14:10
• I'm afraid I can't control the coefficients. I want to prove that for any values of the coefficients (which will assure 2x real and 4x complex solutions), the distance between the real solutions is lower than a threshold. – Manda Mar 22 '18 at 15:44